Problem: Determine how many solutions exist for the system of equations. ${-4x+y = 7}$ ${12x+3y = -15}$
Answer: Convert both equations to slope-intercept form: ${-4x+y = 7}$ $-4x{+4x} + y = 7{+4x}$ $y = 7+4x$ ${y = 4x+7}$ ${12x+3y = -15}$ $12x{-12x} + 3y = -15{-12x}$ $3y = -15-12x$ $y = -5-4x$ ${y = -4x-5}$ Just by looking at both equations in slope-intercept form, what can you determine? ${y = 4x+7}$ ${y = -4x-5}$ The linear equations have different slopes. ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ ${1}$ ${2}$ ${3}$ ${4}$ ${5}$ ${6}$ ${7}$ ${8}$ ${9}$ ${\llap{-}2}$ ${\llap{-}3}$ ${\llap{-}4}$ ${\llap{-}5}$ ${\llap{-}6}$ ${\llap{-}7}$ ${\llap{-}8}$ ${\llap{-}9}$ When two equations have different slopes, the lines will intersect once with one solution.